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An Emergency Department Patient Flow Model Based on Queueing Theory Principles

2013; Wiley; Volume: 20; Issue: 9 Linguagem: Inglês

10.1111/acem.12215

1553-2712

Jennifer L. Wiler, Ehsan Bolandifar, Richard T. Griffey, R Poirier, Tava Lennon Olsen,

Hospital Admissions and Outcomes

Academic Emergency MedicineVolume 20, Issue 9 p. 939-946 Research Methods and StatisticsFree Access An Emergency Department Patient Flow Model Based on Queueing Theory Principles Jennifer L. Wiler MD, MBA, Corresponding Author Jennifer L. Wiler MD, MBA Department of Emergency Medicine, University of Colorado School of Medicine, Aurora, CO Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MOAddress for correspondence and reprints: Jennifer L. Wiler, MD, MBA; e-mail: [email protected].Search for more papers by this authorEhsan Bolandifar PhD, Ehsan Bolandifar PhD Department of Decision Science and Managerial Economics, Chinese University of Hong Kong, Shatin, NT, Hong KongSearch for more papers by this authorRichard T. Griffey MD, MPH, Richard T. Griffey MD, MPH Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MOSearch for more papers by this authorRobert F. Poirier MD, Robert F. Poirier MD Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MOSearch for more papers by this authorTava Olsen PhD, Tava Olsen PhD Department of Information Systems and Operations Management, University of Auckland, Auckland, New ZealandSearch for more papers by this author Jennifer L. Wiler MD, MBA, Corresponding Author Jennifer L. Wiler MD, MBA Department of Emergency Medicine, University of Colorado School of Medicine, Aurora, CO Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MOAddress for correspondence and reprints: Jennifer L. Wiler, MD, MBA; e-mail: [email protected].Search for more papers by this authorEhsan Bolandifar PhD, Ehsan Bolandifar PhD Department of Decision Science and Managerial Economics, Chinese University of Hong Kong, Shatin, NT, Hong KongSearch for more papers by this authorRichard T. Griffey MD, MPH, Richard T. Griffey MD, MPH Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MOSearch for more papers by this authorRobert F. Poirier MD, Robert F. Poirier MD Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MOSearch for more papers by this authorTava Olsen PhD, Tava Olsen PhD Department of Information Systems and Operations Management, University of Auckland, Auckland, New ZealandSearch for more papers by this author First published: 16 September 2013 https://doi.org/10.1111/acem.12215Citations: 47 Dr. Griffey, who is an Associate Editor for this journal, had no role in the peer review process or publication decision. The authors have no relevant financial information or potential conflicts of interest to disclose. AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstracten Objectives The objective was to derive and validate a novel queuing theory–based model that predicts the effect of various patient crowding scenarios on patient left without being seen (LWBS) rates. Methods Retrospective data were collected from all patient presentations to triage at an urban, academic, adult-only emergency department (ED) with 87,705 visits in calendar year 2008. Data from specific time windows during the day were divided into derivation and validation sets based on odd or even days. Patient records with incomplete time data were excluded. With an established call center queueing model, input variables were modified to adapt this model to the ED setting, while satisfying the underlying assumptions of queueing theory. The primary aim was the derivation and validation of an ED flow model. Chi-square and Student's t-tests were used for model derivation and validation. The secondary aim was estimating the effect of varying ED patient arrival and boarding scenarios on LWBS rates using this model. Results The assumption of stationarity of the model was validated for three time periods (peak arrival rate = 10:00 a.m. to 12:00 p.m.; a moderate arrival rate = 8:00 a.m. to 10:00 a.m.; and lowest arrival rate = 4:00 a.m. to 6:00 a.m.) and for different days of the week and month. Between 10:00 a.m. and 12:00 p.m., defined as the primary study period representing peak arrivals, 3.9% (n = 4,038) of patients LWBS. Using the derived model, the predicted LWBS rate was 4%. LWBS rates increased as the rate of ED patient arrivals, treatment times, and ED boarding times increased. A 10% increase in hourly ED patient arrivals from the observed average arrival rate increased the predicted LWBS rate to 10.8%; a 10% decrease in hourly ED patient arrivals from the observed average arrival rate predicted a 1.6% LWBS rate. A 30-minute decrease in treatment time from the observed average treatment time predicted a 1.4% LWBS. A 1% increase in patient arrivals has the same effect on LWBS rates as a 1% increase in treatment time. Reducing boarding times by 10% is expected to reduce LWBS rates by approximately 0.8%. Conclusions This novel queuing theory–based model predicts the effect of patient arrivals, treatment time, and ED boarding on the rate of patients who LWBS at one institution. More studies are needed to validate this model across other institutions. Resumenes Un Modelo de Flujo de Pacientes a través del Servicio de Urgencias Basado en los Principios de la Teoría de Colas Objetivos Derivar y validar un modelo novedoso basado en la teoría de colas que predice el impacto de varios escenarios de saturación de pacientes en el porcentaje de pacientes dados de alta sin ser atendidos (PDASSA). Metodología Se recogieron retrospectivamente los datos de todas las visitas de pacientes al triaje de un servicio de urgencias (SU) sólo para adultos, urbano y universitario, que recibió 87.705 visitas el año 2008. Los datos de las franjas horarias diarias predefinidas se dividieron en grupos de derivación y validación basados en días pares e impares. Se excluyeron las historias de pacientes con datos incompletos de tiempo. Usando un modelo de colas establecido de centralita de llamadas, se modificaron las variables de entrada para adaptar este modelo al escenario del SU, mientras se satisfacían los supuestos subyacentes de la teoría de colas. El objetivo principal fue la derivación y la validación de un modelo de flujo del SU. Se utilizaron los test de la t de Student y de ji-cuadrado para la derivación y la validación del modelo. El objetivo secundario fue estimar el impacto de la variación en la llegada de pacientes al SU y los escenarios de tiempo de espera en el porcentaje de PDASSA usando este modelo. Resultados El supuesto estacionario del modelo fue validado para tres periodos de tiempo, el porcentaje de pico de llegada (10:00am a 12:00 p.m.), el porcentaje de llegada moderada (8:00 a.m. a 10:00 a.m.) y el porcentaje de menos llegada (4:00 a.m. a 6:00 a.m.) para diferentes días de la semana y mes. Entre las 10:00 a.m. y 12:00 p.m., definido como el principal periodo del estudio que representa el pico de llegadas, el 3,9% (n = 4.038) de casos fueron PDASSA. Mediante el modelo derivado, el porcentaje de PDASSA declarado fue de un 4%. Los porcentajes se incrementaron a la vez que se incrementaba el porcentaje de llegadas de pacientes al SU, los tiempos de tratamiento y los tiempos de espera de cama del SU. Un incremento de un 10% en las llegadas horarias de pacientes al SU por encima del porcentaje de llegadas medio observado incrementó el porcentaje de PDASSA predicho al 10,8%; un 10% de descenso en las llegadas horarias de pacientes al SU por debajo del porcentaje medio de llegadas observadas predijo un porcentaje de PDASSA de un 1,6%. Un descenso de 30 minutos en el tiempo de tratamiento por debajo del tiempo de tratamiento medio observado predijo un 1,4% de PDASSA. Un 1% de incremento en la llegada de pacientes tiene el mismo efecto en los porcentajes de PDASSA que un incremento del 1% en el tiempo de tratamiento. La reducción de los tiempos de espera de cama en un 10% reduciría los porcentajes de PDASSA aproximadamente en un 0,8%. Conclusiones Este novedoso modelo basado en la teoría de colas predice el impacto de las llegadas de los pacientes, el tiempo de tratamiento y el tiempo de espera de cama del SU sobre el porcentaje de PDASSA en una institución. Son necesarios más estudios para validar este modelo en otras instituciones. Emergency department (ED) crowding has received considerable national attention in recent years.1, 2 From 1996 to 2006, the annual number of ED visits in the United States increased nearly 32%, from 90.3 million to 119.2 million, while the number of hospital EDs decreased nearly 5%.3 The holding in the ED of patients admitted to the hospital (ED boarding) has also been noted to be a growing problem and is a large contributor to ED crowding.4 ED crowding is known to increase patient wait times.4-6 From 1997 to 2004 and then to 2006, the median wait time to see a physician increased from 38 to 47 to 56 minutes, an increase of 36%.7 As wait times increase, the rate of patients who leave without being seen (LWBS) also increases.8-14 ED patients who leave with being evaluated by a physician are at risk for poorer health outcomes,10, 13, 15 represent a source of lost revenue for hospitals,16 and decrease patient satisfaction.10 For these and other reasons, research efforts have been directed at predicting ED patient load volumes to inform real-time operational interventions with the objectives of managing surge conditions, crowding, and wait times. Various approaches to forecast ED patient volumes have been proposed and studied.17-22 However, no approach has been demonstrated to reliably define crowded conditions when applied to diverse practice settings23 or to reliably outperform simple indices such as bed occupancy rate.24 Some have proposed queueing theory as a logical next step in modeling ED census and crowding.2, 25, 26 Queueing theory makes basic assumptions about a system to create mathematical equations that describe system flow when there are variable inputs and fixed resources. Derived in large part from the telecommunications industry, this methodology has a potentially useful application to the ED setting, where patient flow modeling could predict patient waits. Although many service industry–related queuing models exist, application to the ED setting has been limited.27-37 Previous work has considered portable radiology workflow,28 meeting specific ED disposition time targets,31 ED staffing,34 hospital bed resource allocation, priorities for admission,35 revenue losses from LWBS patients,36 fast track,37 and prehospital operations.38-41 None has been used to predict patients who LWBS based on wait time tolerance and ED crowding. Therefore, our primary aim was the derivation and validation of an ED flow model based on the novel modification of a queueing model commonly used in the call center industry. The secondary aim was estimating the effect of varying ED patient arrival and boarding scenarios on LWBS rates using this model. Methods Study Design The established queueing model M/GI/r/s + GI42 describes customer reneging (leaving system before completing evaluation) after prolonged call center wait times (see Table 1 for explanation of terms). It is the most accurate available queueing model that describes highly variable systems where multiple customers (patients) are served (treated) in parallel, while allowing customer reneging, multiple servers, and a finite waiting room volume. Using this established call center queueing model, we modified model input variables to adapt this model to the ED setting (Table 1), while satisfying the underlying assumptions of queueing theory. This study was approved by the institutional review board. Table 1. Summary of Queueing Model M/M/r/s + M(n) Inputs Queueing Model Term Call Center Application Modification for ED System First M Interarrival times between calls to the system assumed to follow an exponential distribution.a Time between ED patient arrivals. Second M Time speaking to call center agent follows an exponential time distribution. Treatment time (including boarding). r Number of agents available to take calls. Total ED treatment space (bed) capacity. s Maximum capacity of call center to accommodate calls. Waiting area capacity (i.e., maximum number of patients who will wait for evaluation). M(n) Caller waiting time tolerance distribution approximated by an exponential distribution as a function of total number of callers waiting. Patient waiting time tolerance to see provider calculated from a Weibull distribution.46 a Arrivals occur with a known average rate and the number of arrivals in some fixed time period are independent of the number of arrivals in a nonoverlapping time period.20 Study Setting and Population Data for all patients who registered at triage during calendar year 2008 were collected from an urban academic adult-only ED with an annual volume of 87,705 patients, using the institution's electronic medical record system (Healthmatics ED, Allscripts, Chicago, IL). Visits with missing or incomplete operational patient flow metric data were excluded (n = 647). Time stamps for operational patient flow metrics obtained were: 1) patient arrival time; 2) Emergency Severity Index triage acuity score; 3) ED bed placement time; 4) patient time to LWBS (this time stamp occurs when patient is called from waiting area to be placed in ED patient treatment area, but does not respond; most who leave do not notify ED staff; the institution's practice is to call the patient three different times, and record a time stamp for each attempt, and the patient is recorded as a LWBS after third attempt; the first of these attempt time stamps was used for our analysis, as an approximation of the LWBS time); 5) total treatment time (time interval from patient sign-in at triage to time patient is admitted, discharged, transferred to another facility, leaves against medical advice, or expires); and 6) ED boarding time (defined as starting 2 hours after decision to admit was documented). Study Protocol A complete description of the model derivation and specification are provided in Data Supplement S1 (available as supporting information in the online version of this paper). Briefly, as in the article by Whitt,42 we approximate the M/GI/r/s +GI model with the established queueing model M/M/r/s + M(n) with inputs adapted to describe ED patient flow (Table 1). Assumptions required to derive and specify the M/M/r/s + M(n) model are described in Table 2. During the primary study period, arrivals between 10:00 a.m. and 12:00 p.m. were adequately stationary (i.e., stable patient flow) and represented 12.3% of ED patient daily arrivals. We have chosen this time period as our key period of study because it has the largest arrival rate of the day for our studied institution, as is demonstrated in Supplement Figure 1. To evaluate stationarity of the arrival rate during the study period we use a method similar to that of Brown et al.43 who tested stationarity of arrivals in their call center model. We tested for stationarity of arrivals between 10:00 a.m. to 12:00 p.m., and at a 1% significance level, we found no reason to reject the stationarity assumption in 96% of the observed days. Table 2. Required Assumptions and Inputs of a Queueing Theory Modeling Required Assumptions and Inputs Description Methodology Application to Study ED Stationarity Stable flow (i.e., rates of arrivals and departures are constant). Models typically use a relatively stationary period for service demand (e.g., lunchtime for fast food queueing analysis). Arrivals between 10:00 a.m. and 12:00 p.m. were adequately stationary for analysis. Interarrival and service probability distribution Rate of arrivals and service delivery. Continuous standard distribution models (e.g., lognormal, logistic, Student's t, Weibull, beta, etc) are tested to determine the best fit. Daily patient interarrival and ED length of stay exponential distributions were approximated using standard maximum likelihood estimation methodology.47 Prioritization Order in which customers are serviced. Many (e.g., first-come first-serve, first-come first-out, last-in first-out) Requires collapse of patient acuity segmentation (e.g., emergency severity index classification) into a single acuity class in order to be mathematically tenable. Server Fixed capacity to service customers. Servers can be providers (e.g., bank teller) or space (e.g., number of ED beds). Because fast track and observation areas were used as needed in times of crowding, a calculated "effective number of beds" was used. Queue capacity Describes how long the line to receive services is or can be. Once this "capacity" is saturated, all patients are diverted out of system. Waiting area capacity expanded to accommodate walk-in patients but finite capacity set to model ambulance diversion. Waiting time tolerance Assumed each customer has a wait time tolerance that is independent of others waiting in the queue. A general distribution is allowed for tolerance. The Weibull distribution was applied to describe actual tolerance. Figure 1Open in figure viewerPowerPoint Weibull distribution of patients' estimated waiting time tolerance. In addition to these observations for stationarity during each day, we have tested stationarity of arrivals among different days. To do this, we divided all days into 12 bins (each bin represents a month) to estimate standard deviation of arrival rate in this period of time. We used a Student's t distribution with 11 degrees of freedom to find 95% confidence intervals (CIs) for arrival rates, and we found that all arrival rates lie within this 95% CI. Therefore, there is no reason to reject stationarity of arrival rates among different months of the year. To further test stationarity, we also divided our dataset into seven bins (each bin represents a day of a week). To validate our model we selected the 2-hour period with a moderate patient arrival rate (8:00 a.m. to 10:00 a.m.) and the 2-hour period with the lowest arrival rate (4:00 a.m. to 6:00 a.m.). Given these observations, we believe that our defined study period is quite stationary at the daily level within the prescribed time windows; therefore, we can apply our queueing tools to analyze patients' LWBS behavior in this period. However, as queueing models are heavily reliant on stationarity assumptions, readers are cautioned both to test for stationarity before applying the models of this article in their settings and to realize that the results apply narrowly for the modeled stationary period. The number of ED treatment spaces at the study institution varied depending on the time of day, because the fast track area was closed in the late evenings, and a 12-bed observation unit with one additional provider could be used for ED boarding patients as a surge overflow for additional bed capacity when needed. To account for this, a fixed bed capacity r was approximated for the model and designated as the effective number of beds. Thus, the number of "effective beds" was defined as a fixed bed capacity using data from odd days for the time period of 10:00 a.m. to 12:00 p.m. (the same key time period used for the stationarity assumption). The term "s" describes the total number of patients who will wait for evaluation. It was assumed that once this "capacity" is saturated, all patients are diverted out of the system. Because this does not account for walk-in patients, the modeled waiting capacity was expanded to appropriately describe the study ED walk-in volume. This is consistent with previous studies that have also modeled ambulance diversion and excessive patient waits using a fixed waiting area capacity estimate.44, 45 Data Analysis We used a Weibull probability distribution46 to describe patient wait time tolerance (Figure 1). To predict LWBS rates for the system, we modified the call center M/GI/r/s + GI queueing model using a methodology described by Whitt.42 Whitt developed an algorithm to rapidly compute approximations for all of the standard steady-state performance measures in the basic call center queueing model M/GI/s/r + GI, which has a Poisson arrival process, independent and identically distributed service times with a general distribution, s servers, r extra waiting spaces, and customer abandonment times with a general distribution. Simulation experiments by Whitt showed that the approximation is quite accurate to predict abandonment in call center customers.42 In this article, we have applied this algorithm to determine the LWBS rate from our studied ED after finding and calibrating the best fit distributions for interarrival, treatment, and tolerance times. We used maximum likelihood estimation47 and the expectation maximization48 algorithms to identify distributional parameters of waiting time tolerance and total "ED bed occupancy" (sum of ED treatment and boarding time) to fully specify the model. Data were divided into derivation (odd dates) and validation (even dates) sets. We divided the derivation data set into 15 time-based bins spread across time (pattern of bin 1 = days 1, 31, 61, 91, …; bin 2 = days 3, 33, 63, 93, …; bin 15 = 29, 59, 89, 119, …, etc. repeated through 365 days) and calculated the LWBS rate for each bin. We used the derivation set to find the standard deviation of the observed LWBS rate. We then used the estimated SD to find a 95% CI for the predicted LWBS rate of the validation set based on a Student's t-test with 14 degrees of freedom, which is 95% CI = 0.41 to 6.92. To validate our model, we used the validation data set to compute input parameters of our model including average arrival rate and service and boarding times. Substituting for these parameters in our model we computed predicted LWBS rate for the validation data set. Because the predicted LWBS (2.75%) lies in the computed CI, we did not find any evidence to reject validity of our model. We also performed a secondary validation using data from weekends, determined a priori to be a more homogeneous time period. Similar to the previous validation test we computed a 95% CI for the predicted LWBS rate of the validation set based on a Student's t-test with 14 degrees of freedom: 95% CI = 0 to 4.59 (which is quite different from the previous validation test since weekends are a more homogeneous period). Again, our predicted LWBS rate (3%) from our model lies in this CI, which provides no evidence to reject validity of our model. We determined the effect of ED patient arrival rates, treatment times, and boarding on LWBS rates using the validated model. Results To validate our model we selected a 2-hour period with a moderate patient arrival rate (8:00 a.m. to 10:00 a.m., total arrival of 8,304 patients, 9.5% of all ED visits) and a 2-hour period with the lowest arrival rate (4:00 a.m. to 6:00 a.m., total arrival of 4,239 patients, 4.8% of all ED visits). As we mentioned earlier, the test described by Brown et al.43 demonstrated that arrivals in these periods were stationary, so we could apply our model to predict LWBS rate in these periods. Using the same methodology for each of these periods, we found that for the time period of 8:00 a.m. to 10:00 a.m., the 95% CI for LWBS rate on even days is 95% CI = 0 to 5.25. Since our model prediction is 1%, which lies in this interval, there is no evidence to reject our model prediction. Similarly for the period of 4:00 a.m. to 6:00 a.m., the CI for the LWBS rate on even days is 95% CI = 0 to 5.65. Since our model prediction (2%) lies within this interval, again there is no evidence to reject our model prediction. We also performed a secondary validation test using data from weekends. We used arrivals on Saturday as the derivation set to calibrate our model and compute 95% CI for the predicted LWBS rate of the validation set (which is arrivals on Sunday) based on a Student's t-test with 14 degrees of freedom. For the period of 8:00 a.m. -10:00 a.m., the 95% CI for LWBS rate on Sundays is 0 to 4.25. Since our model prediction is 2%, there is no evidence to reject our model prediction. Similarly, for the period of 4:00 a.m. to 6:00 a.m., the LWBS rate on even days is 95% CI = 0.15 to 6.65, which includes our model prediction of LWBS rate on Sundays, so there is no evidence to reject our model prediction. Estimating the Effect of ED Patient Arrivals and Boarding on LWBS Rates Using a Novel Model The effect of varying ED patient arrivals and boarding on LWBS rates was determined using the key stationary time period, defined as being between 10:00 a.m. and 12:00 p.m. (i.e., peak patient arrival time; see Table 2 for definition), when 4.1% of patients (n = 418) LWBS by a provider. The mean (±SD) wait time tolerance for the system (i.e., actual study ED population) during this time period was 10.68 (±7.76) hours. The actual versus model-predicted LWBS rates are presented in Table 3. The average wait time to see a provider using the model was 85 minutes, which was very close to the actual average wait time of 89 minutes. Table 3. Actual Versus Model Predicted LWBS Rates for Key Study Period (10:00 a.m. to 12:00 p.m.) Quarter Observed LWBS Modeled LWBS Lower 95% CI Interval Upper 95% CI interval 1 6.7 9 3.7 9.7 2 3.3 2.1 0.3 6.3 3 3.5 4.8 0.5 6.5 4 2.6 2.9 0 5.6 Data are reported as percentages. LWBS = left without being seen. The effect of ED arrivals by hour on LWBS rates is provided in Figure 2. A 10% increase in hourly ED patient arrivals from the observed average arrival rate predicted a 10.74% LWBS rate. A 10% decrease in hourly arrivals from the observed average arrival rate predicted a 1.6% LWBS rate. The duration of treatment time also influences the rate of LWBS (Figure 3). Specifically, a 30-minute decrease in treatment time from the observed average treatment time predicts a 1.4% LWBS rate. It was observed that a 1% increase in the rate of ED patient arrivals has the same effect on LWBS rates as a 1% increase in treatment time. Figure 2Open in figure viewerPowerPoint Effect of ED arrivals by hour on LWBS rates. LWBS = left without being seen. Figure 3Open in figure viewerPowerPoint Duration of treatment time influences LWBS rates. LWBS = left without being seen. Reducing the number of admitted patients in the ED who are waiting for an inpatient bed i.e., "boarding", notably reduces LWBS rates (Figure 4). Reducing boarding times by 10% is expected to reduce LWBS rates by approximately 0.8%, with a 50% reduction expected to decrease the LWBS rate to 1.5% (from the actual 4%) in the study ED. Figure 4Open in figure viewerPowerPoint Reducing the number of admitted patients in the ED who are waiting for a inpatient beds ("boarding") reduces LWBS rates. LWBS = left without being seen. Discussion Crowding is known to prompt patients to leave EDs without being seen by providers. Numerous studies have catalogued the characteristics of patients who LWBS, but none has described a mathematical prediction tool to help inform ED operations.8-15 Queueing models lend themselves well to describing the ED environment because they allow for the application of simple equations to model patient flow. In most ED applications these equations can be easily input into a spreadsheet. At the most basic level, queueing systems consist of four components: arrivals, servers, service principles (described as the "queueing discipline" or rules as to whom a server serves next), and the flow or routing of the customer or item through the system. These models then describe the effect of varying demand on wait times, waiting tolerance, capacity, and utilization metrics. A handful of queueing models have been designed to describe ED patient flow29, 31 and predict demand in the ED.27, 30 However, none has been ideal to describe the complex ED environment, nor describe the effect of patient demand on LWBS rates. Using our calibrated M/M/r/s + M(n) model (Table 1), we found that ED LWBS rates climb in a predictable and exponential way as the rate of ED patient arrivals increases. This is not surprising, since as more patients arrive per hour, the queue to be served (i.e., bed placement in our model) grows, resulting in longer wait times for patients. Strategies that obviate the need for bed placement (e.g., treat and release by a provider in triage) would be expected to have a positive effect on LWBS rates, but these were not modeled in our study. The overall mean waiting time tolerance in our patient population was nearly 11 hours, and indeed most patients (93% of all arrivals to ED) stayed for treatment rather than LWBS. Identifying the patient wait time tolerance distribution(s) was challenging because the study data necessarily censored for those who remained in queue and were ultimately evaluated by a physician. Our ED is located in an urban center and had notable issues with crowding during the study period. It is not known if the waiting time tolerance of patients at the study institution were affected by the regular crowded conditions or if the wait time tolerance mirrors that of other ED populations. However, this is the first description of patient waiting time to